Hello Drew.
Which version of Music does your theory relate to. Is it the harmonic structures of Pythagoras, based around the harmonies of the 5th, which dont extrapolate to a coherent octave, or, the mathematically derived equal temperament octave, which doesnt have a single harmonic in the octave.
It is the Pythagorean one but there is coherence due to
noncommutativity as Alain Connes explains. So you are correct, the
octaves are not a symmetric equivalence.
The response back:Thanks for the explanation. I know Alain Connes work and he is mathematically based on the principles as per equal temperament.
That was why I asked the question.
The mathematical divisions of Equal temperament derive a functional scale, but the Pythagorean divisions are harmonically coherent only to limited notes within the scale, and even then not accurate beyond the precise values of the 4th and 5th scale tones.
It does not seem possible to have, and combine, the two systems as the Pythagorean s based on the reference of a single tone to one other, and simply do not make coherent music, and the coherent music, as per Connes and equal temperament do not have single harmonic element within them.
It would seem that your very clever and interesting article ignores this fundamental, and seemingly unsolvable dichotomy.
To rephrase my question, is it Connes or Pythagoras.
I
guess we disagree about what Connes thinks about music theory. haha. I
have a new article about to be published that gives the details from
Connes. The quotes from Connes and Michio Durdevich quoting Connes
confirms my claim - with further details of course.
Still I definitely am not trying to change your views. Just so you know.
thanks for sharing them!
drew
Still I definitely am not trying to change your views. Just so you know.
thanks for sharing them!
drew
...........................................................................
So this person was clearly mad because I could feel the energy. In fact I'm pretty sure they were motivated to contact me for other reasons that caused them to be mad - and not really the topic at hand. haha.
To explain in more detail my response: Alain Connes clearly explains that the logarithmic symmetric music scale is precisely the "scale" but it's not the same as the two note chords. So the scale is derived from the two note noncommutative phase chord dynamic.
Connes then states that the two note chord is noncommutative so that there is more than one two note chord for each zero point in space - hence the "triple spectral."
I should have included another quote from Connes to better explain but essentially the noncommutative chord is empirically true but can not be allowed for the symmetric logarithmic scale of spacetime. And it is precisely that hidden, "behind the scenes" dynamic of nonlocal noncommutative time-frequency that CREATES the zero point in space as the "symmetric" scale.
So contrary to this person's claim - the Pythagorean and Connes view are not diametrically opposed as Michio Durdevich also makes clear.
So Connes calls the triple spectral as based on the noncommutative "double quotient" of (2, 3, infinity). But the key here is that the infinity is relative to the zero point in space and that is what creates the zero point in space.
So you have the octave as one chord and then the 3/2 as the other chord and 2/3 as the third chord. The 2/3 is not allowed since it has to be doubled back into the same octave to create the zero point in space.
This is why in the Harmonic Series the fundamental frequency is either the DOUBLE octave - thereby hiding the conversion of 2/3 into 4/3 OR it already assumes the zero point in space created due to the irrational alogon square root equation.
The fact that this mathematician was mad at me is precisely why this research is so important - because there is a crucial "mathematical structure" that Alain Connes is revealing and as he states - it is considered strange and a nuisance to almost all scientists since almost all science is based on the symmetric algebraic geometry that first originated from the equal-tempered music scale.
Aristotle by the way was against the Zero as a "negative infinity" that was a materialistic geometry of alogon.
In this pattern, there is only one-half of a wave within the length of the string. This is the case for the first harmonic or fundamental frequency of a guitar string.
Notice how the author just switched around "wave" with "string" - wave of WHAT? How can a wave be twice as long as the string and yet the sound is supposedly produced by the string? And then later on it's the wave that is one-half the length of the string! Nice "bait and switch" logic there!
So what is very hilarious is that this is the Double Octave or Greater Perfect System from music theory that covered up the noncommutative phase!! This is what Philolaus used to CREATE the Greek Miracle foundation of science. But the physicists never explain this of course as they have no idea.
Alain Connes is explaining that the noncommutative musical chords are not just in reference to each other - but rather they are nonlocality as the foundation of reality - hence the "infinity" part of the 2/3 and 3/2 music chord.
So the obvious question in the Harmonic Series is WHY does the "fundamental frequency" HAVE to start with 2/1 and not just 1/1? It's the FIRST harmonic yet it has to be 2/1 !! The answer again is to cover up the noncommutative phase truth of reality.
So how do they get away with this? By converting the Double Octave or 2/1 wavelength into a geometric symbol aka the (2x) value in the "Doe a Deer" scale. So that the "octave" is inherently actually already a SQUARE of geometry! That's not how frequency-time resonates as infinity, as Connes points out. haha.
So if the "first harmonic" is a 2/1 wavelength as a double octave then this Lie started with Philolaus.
Physics covers it up by LYING about music theory. Mathematics then just relies on the physics and vice versa - until someone like Alain Connes comes along. haha.
So the key to understand Connes is that he's relying on the "inner automorphisms" of the Dirac Operator as Matrix math. That means it's the noncommutative diagonal.
So the musical chord is the "line element" and "the coordinates on a space" is the musical scale.
Here's the Connes' quote for you:
As Connes [8] emphasizes
As Connes [8] emphasizes
“It is precisely this lack of commutativity between the lineAnd as it relates to music theory, Connes again:
element and the coordinates on a space [between ds and the
a ∈ A] that will provide the measurement of distance.”
"It is precisely the irrationality of log(3)/ log(2) which is responsible for the noncommutative nature of the quotient corresponding to the three places {2, 3,∞}. "
I'll dig up those Connes quotes...
Alain
Connes: "The point of noncommutative geometry is NOT to extend classical
concepts and so on, to this situation,...it's more complicated than
that. The point is there will be totally NEW phenomenas in the
noncommutative case which will have NO commutative counterpart. This is
really the important point. And if we look at the hierarchy of the
levels of understanding of we have of an ordinary space...I mean - so
you see - so something very trivial is the following - when you look at
the space you can look at it from various points of view. The closest
point of view is Measure Theory, then you have Topology, then you have
Differential geometry, by this I mean the soft part, there are
currents... Finally you have geometry per se which is the metric
aspect....EVEN at the measure theory level, there is a completely NEW
phenomenon, which was really the starting point of the whole story,
which is the following very striking fact. So I will not talk about
spaces - because I will convey all the structure to the corresponding
algebra. There is a really striking fact which has probably not passed
the mathematical community, which is the following one: IN the Measure
Theory - which means it's only a certain kind of algebra which is
considered, you are NOT allowed to stay what is a continuous function,
it's a certain kind of algebra... An algebra A as a God-Given time
evolution - so it moves, it rotates, it's a very strange thing. There is
an absolutely God-Given morphism from the real line of the Group to the
Group OUT of the algebra A which is the Quotient of the
automorphisms...and this Group, roughly speaking, what it does is that
it exchanges Left and Right... A very striking feature of noncommutative
geometry from the Measure Theory point of view....standard music
suggests dealing with new shapes which are quantum such as the quantum
2-spheres....On the other hand the stretching of geometric thinking
imposed by passing to noncommutative spaces forces one to rethink about
most of our familiar notions. ...And it could be formalized by music….I
think we might succeed in this way to educate the human mind to deal
with polyphonic situations in which several voices coexist, in which
several states coexist, whereas our ordinary logical allows room for
only one. Finally, we come back to the problem of adaptation, which has
to be resolved in order for us to understand quantum correlation and
interrelation which we discussed earlier, and which are fundamentally
schizoid in nature. It is clear that logic will evolve in parallel with
the development of quantum computers, just as it evolved with computer
science. That will no doubt enable us to cross new borders and to better
integrate the mathematical formalism of the quantum world into our
metaphysical system.... When Riemann wrote his essay on the foundations
of geometry, he was incredibly careful. He said his ideas might not
apply in the very small. Why? He said that the notion of a solid body of
a ray of light doesn't make sense in the very small. So he was
incredibly smart. His idea, I have never been able to understand his
intuition...But however he wrote down explicitly that the geometry of
space, of spacetime, should be encapsulated, should be given by the
forces which hold the space together. Now it turns out this is exactly
what we give here...One day I understood the following: That we are born
in quantum mechanics. We can not deny that... Quantum mechanics has
been verified. The superposition principle has been verified. The spin
system is really a sphere. This has been verified. This has been checked
so many times. That we can not say that Nature is classical. No. Nature
is quantum. Nature is very quantum. From this quantum stuff, we have to
understand our vision, our very classical, because of natural selection
way of seeing things can emerge. It's very very difficult of course.
...Why should Nature require some noncommutativity for the algebra? This
is very strange. For most people noncommutativity is a nuisance. You
see because all of algebraic geometry is done with commutative
variables. Let me try to convince you again, that this is a misgiving.
OK?....Our view of the spacetime is only an approximation, not the
finite points, it's not good for inflation. But the inverse space of
spinors is finite dimensional. Their spectrum is SO DENSE that it
appears continuous but it is not continuous.... It is only because one
drops commutativity that variables with a continuous range can coexist
with variables with a countable range....What is a parameter? The
parameter is time...If you stay in the classical world, you can not have
a good set up for variables. Because variables with a continuous range
can not coexist with variables of discrete range. When you think more,
you find out there is a perfect answer. And this answer is coming from
quantum mechanics....The real variability in the world is exactly is
where are you in the spectrum [frequency] of this variable or operator.
And what is quite amazing is that in this work that I did at the very
beginning of my mathematical studies, the amazing fact is that exactly
time is emerging from the noncommutivity. You think that these variables
do not commute, first of all it is that they don't commute so you can
have the discrete variable that coexists with the continuous variable.
What you find out after awhile is that the origin of time is probably
quantum mechanical and its coming from the fact that thanks to
noncommutativity ONLY that one can write the time evolution of a system,
in temperature, in heat bath, the time evolution is really coming from
the noncommutativity of the variables....You really are in a different
world, then the world of geometry, which we all like because we all like
to draw pictures and think in a geometric manner. So what I am going to
explain is a very strange way to think about geometry, from this point
of view, which is quite different from drawing on the blackboard...I
will start by asking an extremely simple question, which of course has a
geometrical origin. I don't think there can be a simpler question.
Where are we?....The mathematical question, what we want, to say where
we are and this has two parts: What is our universe? What is the
geometric space in which we are? And in which point in this universe we
are. We can not answer the 2nd question without answering the first
question, of course....You have to be able to tell the geometric space
in an invariant manner....These invariants are refinements of the idea
of the diameter. The inverse of the diameter of the space is related to
the first Eigenoperator, capturing the vibrations of the space; the way
you can hear the music of shapes...which would be its scale in the
musical sense; this shape will have a certain number of notes, these
notes will be given by the frequency and form the basic scale, at which
the geometric object is vibrating....The scale of a geometric shape is
actually not enough.... However what emerges, if you know not only the
various frequencies but also the chords, and the point will correspond
to the chords. Then you know the complete thing....It's a rather
delicate thing....There is a very strange mathematical fact...If you
take manifolds of the same dimension, which are extremely
different...the inverse space of the spinor doesn't distinguish between
two manifolds. The Dirac Operator itself has a scale, so it's a spectrum
[frequency]. And the only thing you need to know...is the relative
position of the algebra...the Eigenfunctions of the Dirac Operator....a
"universal scaling system," manifests itself in acoustic
systems....There is something even simpler which is what happens with a
single string. If we take the most elementary shape, which is the
interval, what will happen when we make it vibrate, of course with the
end points fixed, it will vibrate in a very extremely simple manner.
Each of these will produce a sound...When you look at the eigenfunctions
of the disk, at first you don't see a shape but when you look at very
higher frequencies you see a parabola. If you want the dimension of the
shape you are looking at, it is by the growth of these eigenvariables.
When talking about a string it's a straight line. When looking at a two
dimensional object you can tell that because the eigenspectrum is a
parabola.... They are isospectral [frequency with the same area], even
though they are geometrically different [not isomorphic]....when you
take the square root of these numbers, they are the same [frequency]
spectrum but they don't have the same chords. There are three types of
notes which are different....What do I mean by possible chords? I mean
now that you have eigenfunctions, coming from the drawing of the disk or
square [triangle, etc.]. If you look at a point and you look at the
eigenfunction, you can look at the value of the eigenfunction at this
point.... The point [zero in space] makes a chord between two notes.
When the value of the two eigenfunctions [2, 3, infinity] will be
non-zero. ...The corresponding eigenfunctions only leave you one of the
two pieces; so if there is is one in the piece, it is zero on the other
piece and if it is non-zero in the piece it is zero there...You
understand the finite invariant which is behind the scenes which is
allowing you to recover the geometry from the spectrum....Our notion of
point will emerge, a correlation of different frequencies...The space
will be given by the scale. The music of the space will be done by the
various chords. It's not enough to give the scale. You also have to give
which chords are possible....The only thing that matters when you have
these sequences are the ratios, the ear is only sensitive to the ratio,
not to the additivity...multiplication by 2 of the frequency and
transposition, normally the simplest way is multiplication by 3...2 to
the power of 19 [524288] is almost 3 to the power of 12 [531441]....You
see what we are after....it should be a shape, it's spectrum looks like
that...We can draw this spectrum...what do you get? It doesn't look at
all like a parabola! It doesn't look at all like a parabola! It doesn't
look at all like a straight line. It goes up exponentially fast...What
is the dimension of this space?...It's much much smaller. It's
zero...It's smaller than any positive.... Musical shape has geometric
dimension zero... You think you are in bad shape because all the shapes
we know ...but this is ignoring the noncommutative work. This is
ignoring quantum groups. There is a beautiful answer to that, which is
the quantum sphere... .There is a quantum sphere with a geometric
dimension of zero...I have made a keyboard [from the quantum
sphere]....This would be a musical instrument that would never get out
of tune....It's purely spectral....The spectrum of the Dirac
Operator...space is not simply a manifold but multiplied by a
noncommutative finite space......It is precisely the irrationality of
log(3)/ log(2) which is responsible for the noncommutative
[complementary opposites as yin/yang] nature of the quotient
corresponding to the three places {2, 3,∞}. The formula is in
sub-space....Geometry would no longer be dependent on coordinates, it
would be spectral...The thing which is very unpleasant in this formula
is the square root...especially for space with a meter....So there is a
solution to this problem of the square root, which was found by Paul
Dirac....It's not really Paul Dirac, it is Hamilton who found it
first...the quaternions is the Dirac Operator....Replace the geometric
space, by the algebra and the line element...for physicists this thing
has a meaning, a propagator for the Dirac Operator. So it's the inverse
of the Dirac Operator.... You don't lose anything. You can recover the
distance from two points, in a different manner....but by sending a wave
from point A to point B with a constraint on the vibration of the wave,
can not vibrate faster than 1; because what I ask is the commutator of
the Dirac Operator is less than 1...It no longer requires that the space
is connected, it works for discrete space. It no longer requires that
the space is commutative, because it works for noncommutative
space....the algebra of coordinates depends very little on the actual
structure and the line element is very important. What's really
important is there interaction [the noncommutative chord]. When you let
them interact in the same space then everything happens....You should
never think of this finite space as being a commutative space. You have
matrices which are given by a noncommutative space...To have a geometry
you need to have an inverse space and a Dirac Operator...The inverse
space of the finite space is 5 dimensional....What emerges is finite
space... a point of the geometric space "X" can be thought of as a
correlation ...which encodes the scalar product at the point between the
eigenfunctions of the Dirac operator associated to various frequencies,
i.e. eigenvalues of the Dirac operator....It's related to mathematics
and related to the fact that there is behind the scene, when I talk
about the Dirac Operator, there is a square root, and this square root,
when you take a square root there is an ambiguity. And the ambiguity
that is there is coming from the spin structure.... We get this formula
by counting the number of the variables of the line element that are
bigger than the Planck Length. We just count and get an integer....
There is a fine structure in spacetime, exactly as there is a fine
structure in spectrals [frequencies]....Geometry is born in quantum
space; it is invariant because it is observer dependent....Our brain is
an incredible ...perceives things in momentum space of the photons we
receive and manufactures a mental picture. Which is geometric. But what I
am telling you is that I think ...that the fundamental thing is
spectral [frequency]....And somehow in order to think we have to do this
enormous Fourier Transform...not for functions but a Fourier Transform
on geometry. By talking about the "music of shapes" is really a fourier
transform of shape and the fact that we have to do it in reverse. This
is a function that the brain does amazingly well, because we think
geometrically....The quantum observables do no commute; the phase space
of a microscopic system is actually a noncommutative space and that is
what is behind the scenes all the time. They way I understand it is that
some physical laws are so robust, is that if I understand it correctly,
there is a marvelous mathematical structure that is underneath the law,
not a value of a number, but a mathematical structure....A fascinating
aspect of music...is that it allows one to develop further one's
perception of the passing of time. This needs to be understood much
better. Why is time passing? Or better: Why do we have the impression
that time is passes? Because we are immersed in the heat bath of the 3K
radiation from the Big Bang?...time emerges from
noncommutativity....What about the relation with music? One finds
quickly that music is best based on the scale (spectrum) which consists
of all positive integer powers qn for the real number q=2 to the 12th∼3
to the 19th. [the 19th root of 3 = 1.05953 and 12th root of 2=1.05946]
Due to the exponential growth of this spectrum, it cannot correspond to a
familiar shape but to an object of dimension less than any strictly
positive number....This means it is a zero dimensional object! But it
has a positive volume!... As explained in the talk, there is a beautiful
space which has the correct spectrum: the quantum sphere of Poddles,
Dabrowski, Sitarz, Brain, Landi et all. ... We experiment in the talk
with this spectrum and show how well suited it is for playing music. The
new geometry which encodes such new spaces, is then introduced in its
spectral form, it is noncommutative geometry, which is then confronted
with physics....Algebra and Music...music is linked to time exactly as
algebra is....So for me, there is an incredible collusion between music,
perceived in this way, and algebra....I believe that this variability
is more fundamental than the passing of time. And that it's behind the
scene, meaning that the passing of time is a corollary of this....it's
exactly the variability of which value you get in the spectrum of the
operator....It's associated with a specific spin structure; You have the
chirality and the charge conjugation operator....Chirality is put in in
your algebraic formalism. You PUT the chirality. It's an operator and
two spinors, so you put it there. ..This is quite tricky. If you want,
what happens...when you write it because,...it's a two by two
matrix....it's a map to the [quantum] two-sphere....This is a totally
anti-symmetric tensor... Fields Medal math professor Alain Connes
(compilation of quotes)
back to the PDF that quotes Connes...
So what that is referring to is "precisely" from Philolaus - as Richard McKirahan points out.
So McKirahan is assuming the "zero" symmetric value of the scale as the fundamental frequency.
(1, 4) = (7, 5)
What is (1, 4)? It is a Perfect Fourth which is the PITCH of C to F as 4/3.
In the present case this means that regardless if you go from string 1 to string 7 via string 4 or string 5, the result is the same: (1, 7) = 0.
So there you have it - the "start" of the commutative diatonic seven note scale is zero.
He says in the foot note that 0 is the "original note" as the reversal of Perfect Fifth and Perfect Fourth.
So he says it's an octave between the LYRE string of 1 and 7.
So this is Connes point - the "line" is isomorphic - the same geometry as the Perfect Fourth in each case (1, 4) and (7, 5) but ALGEBRAICALLY in time they are noncommutative since the "chord" (and not the scale) is C to F as 1, 4 and C' (octave) to G as (7, 5).
No comments:
Post a Comment